Question

The shadow cast by a vertical pillar in alexandria at noon during the summer solstice is found to be 1/8 the height of the pillar. the distance between alexandria and syene is 1/8 earth's radius. is there a geometric connection between these two 1-to-8 ratios?

Answer 1

At noon on summer solstice, the sun is directly north of Alexandria at latitude 23°26' whereas Alexandria has latitude 31°12'.
The difference is 7°44'.

The difference in distance between Syene (Aswan) and Alexandria is 1/8 of the earth's radius, which translates to an angular distance of (1/8) of a radian, or 7.2°.

So the relationship between the two would be the sun and Syene both have (approximately) the same angular distance from Alexandria.